Non-Holonomic RRT & MPC: Path and Trajectory Planning for an Autonomous Cycle Rickshaw
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Non-Holonomic RRT & MPC: Path and Trajectory Planning for an
Autonomous Cycle Rickshaw
Damir Bojadžić∗ , Julian Kunze∗ , Dinko Osmanković† , Mohammadhossein Malmir‡ , Alois Knoll‡
Abstract— This paper presents a novel hierarchical motion
planning approach based on Rapidly-Exploring Random Trees
(RRT) for global planning and Model Predictive Control (MPC)
for local planning. The approach targets a three-wheeled cycle
rickshaw (trishaw) used for autonomous urban transportation
arXiv:2103.06141v1 [cs.RO] 10 Mar 2021
in shared spaces. Due to the nature of the vehicle, the algorithms
had to be adapted in order to adhere to non-holonomic
kinematic constraints using the Kinematic Single-Track Model.
The vehicle is designed to offer transportation for people and
goods in shared environments such as roads, sidewalks, bicycle
lanes but also open spaces that are often occupied by other
traffic participants. Therefore, the algorithm presented in this
paper needs to anticipate and avoid dynamic obstacles, such as
pedestrians or bicycles, but also be fast enough in order to work
in real-time so that it can adapt to changes in the environment.
Our approach uses an RRT variant for global planning that
has been modified for single-track kinematics and improved by Fig. 1: A photo of our target vehicle with main computer
exploiting dead-end nodes. This allows us to compute global
paths in unstructured environments very fast. In a second step,
and retrofitted sensors highlighted.
our MPC-based local planner makes use of the global path to
compute the vehicle’s trajectory while incorporating dynamic In this regard, the vehicle is designed to appropriately
obstacles such as pedestrians and other road users. address passengers and goods transportation in shared urban
Our approach has shown to work both in simulation as environments like parks or bicycle lanes and also larger
well as first real-life tests and can be easily extended for more exhibition sites or factory premises. Rickshaws are far more
sophisticated behaviors. space efficient than cars for single-person transportation.
Index Terms— Autonomous Vehicle Navigation, Rapidly-
Exploring Random Trees (RRT), Model Predictive Control
Combined with the reduced spatial requirements emerging
(MPC), Kinematic Single-Track Model from autonomous operation, this renders large parts of
parking spaces unnecessary, which therefore allows cities to
I. INTRODUCTION become more human-centered instead of vehicle-centered.
In this paper, we want to tackle some of the significant
In the context of autonomous driving, the focus usually challenges to the vehicle’s motion planning that arise from
lies on cars and trucks that can be used for the transportation autonomous driving in shared urban environments. Firstly,
of people and goods [1], [2], [3]. On the other hand, there these environments require the vehicle to drive not only
are also small wheeled mobile robots that are employed in a network of streets, but also in public open spaces.
for industrial intra-factory logistics and package delivery Secondly, sharing spaces with pedestrians, cyclists and other
tasks [4], [5], [6]. While the former type usually operates road users requires the planner to incorporate their behavior
under a set of traffic regulations and on a network of roads as well. Therefore, our approach consists of a two-step
at velocities of sometimes more than 100 km/h, the latter hierarchical planning architecture with an RRT-based path
is often required to just drive at a pedestrian speed and planner and an MPC-based trajectory planner. The planner
in contrast finds itself in more unstructured environments is embedded into a system architecture based upon the
without a clear set of rules. However, our work targets a Robot Operating System (ROS1) [7] and Kubernetes1 . Our
class of vehicles that is situated between a car and a small RRT planner is able to traverse large open environments
mobile robot, both with respect to the environment as well very quickly while taking care of the vehicle’s kinematic
as the level of dynamics. constraints and to provide a global path on a static map
∗ WARP Student Team & Department of Informatics, Techni-
for our vehicle to follow. The MPC planner computes the
cal University of Munich (TUM), Germany @tum.de dynamic obstacles like pedestrians. While in its current state,
† Department of Automatic Control and Electronics, Faculty of Elec-
the MPC planner only avoids obstacles in a naïve manner,
trical Engineering, University of Sarajevo, Bosnia & Herzegovina
dinko.osmankovic@etf.unsa.ba
its architecture is designed to be able to add sophisticated
‡ Chair of Robotics, Artificial Intelligence and Real-time Systems, De- behavior predictions for other road users easily.
partment of Informatics, Technical University of Munich (TUM), Germany
hossein.malmir@tum.de, knoll@mytum.de 1 https://kubernetes.io/II. RELATED WORK Figure 1) where driver seat, handlebar, and pedal-powered
A. Rapidly-Exploring Random Trees drive have been removed. Instead, the rear wheel electric
drive is controlled directly by our computer. Moreover, a
Rapidly-Exploring Random Trees (RRT) count toward
separate steering motor and braking actuator have been
some of the most popular planning algorithms. Many mod-
installed to allow these functions to be computer-operated
ifications and improvements to the algorithm have been
as well.
introduced. One of such is presented in [8], where the au-
thors introduce the Theta∗ -RRT algorithm that hierarchically Besides the vehicle’s proprioceptive sensors for measuring
combines (discrete) any-angle search with (continuous) RRT e.g. accelerations, wheel speed and steering angle, several
motion planning for non-holonomic wheeled robots. The exteroceptive sensors have been added in order to allow
authors show that this variant outperforms its siblings RRT environmental perception and mapping. Figure 1 highlights
[9], A∗ -RRT [10], RRT∗ [11] and A∗ -RRT∗ [12] both in path the main computer and sensors that have been retrofitted to
length and speed. the vehicle. For triggering emergency braking, a 270◦ FOV
Another extension of RRT is presented in [13], where the 2D Lidar has been mounted at the front. Furthermore, data
authors suggest the usage of so-called key configurations for from an RGB camera and a 3D Lidar is fused for object
tackling problems such as moving through narrow passages detection, providing information on static and dynamic ob-
with non-holonomic vehicles. stacles. Finally, several proprioceptive sensors are combined
with a GNSS sensor as well as the 3D Lidar for mapping
B. Model Predictive Control the environment and localizing the vehicle on that map. The
This paper was additionally inspired by the work of [14], navigation stack uses processed obstacle information, map
where the authors utilized a Model Predictive Control (MPC) and vehicle pose to first generate a global path and then
approach for local planning. Moreover, global path cost- to iteratively compute a trajectory for the vehicle to follow.
to-go computation is done via a fast D∗ -lite algorithm, Finally, simple PID controllers for each separate actuator
and a Gaussian smoothing is applied on the precomputed make sure that the trajectory is being correctly executed.
costmap. The authors show that MPC can run on such a Figure 2 gives an overview over the key components that
precomputed map and that MPC is robust enough to adapt are relevant for autonomously operating the vehicle.
to dynamic obstacles that appear at runtime that were not Regarding our software architecture, all components have
part of the original costmap. That way, the dynamic nature been containerized and are orchestrated via K3s Kubernetes3
of the environment is also accounted for. in order to simplify a capsuled development of individual
In addition to the previous research paper, [15] represents components as well as to allow for hot redundancy which
a novel online motion planning approach based on nonlinear is an important feature for safety critical systems [17]. In
MPC. Using non-euclidean rotation groups, the authors have addition, we employ ROS14 to facilitate the communication
formulated an optimization problem that solves local plan- between individual components. Consequently, our naviga-
ning via optimal control. The authors demonstrate that the tion stack is based on the ROS navigation package5 of which
proposed algorithm is able to solve a quasi-optimal parking we use the default map server but implement our own global
maneuver whilst also avoiding dynamic obstacles. and local planner. For mapping and localization, we modified
The power and capability of MPC extends even to rac- the Google Cartographer ROS integration6 to match our
ing scenarios. In [16] the authors propose a Linear Time sensor setup.
Varying Model Predictive Control (LTV-MPC) approach for The next section will go into detail on how the global
achieving minimum lap time in a racing scenario while also and local planner have been designed in order to match our
modeling the ability of the vehicle to drift in corners, thus use-case.
giving it a time advantage. The MPC approach was not
IV. APPROACH
only able to keep up with the high demand and complex
dynamic model of the vehicle, but also maintains stability A. GLOBAL PLANNER
under extreme drifting conditions.
The global planner computes a global path X from
To the best of the authors knowledge, there is no existing
starting state [x0 , y0 , θ0 ] to goal state [xn , yn , θn ] with non-
prior work that suggests using Non-Holonomic RRT for
predefined length n on a given map. The global path is
global planning and online MPC for local planning designed
then defined as an array of individual states as shown in
for a three-wheeled cycle rickshaw. In the following sections,
Equation 1.
the vehicle and system architecture are first introduced
followed by in-depth description of the adapted approach
x0 x1 x2 xn
for path and trajectory planning steps.
X = y0 , y1 , y2 , ..., yn (1)
III. VEHICLE AND SYSTEM ARCHITECTURE θ0 θ1 θ2 θn
Our approach targets a trishaw (three-wheeled cycle
rickshaw) vehicle that has been modified to operate au- 3 https://rancher.com/docs/k3s/latest/en/
4 https://www.ros.org/
tonomously. The vehicle base is a BAYK CRUISER2 (see
5 http://wiki.ros.org/navigation
2 https://bayk.ag/en/cruiser/ 6 http://wiki.ros.org/cartographerPerception
2D Lidar
Static and Emergency
Object Object dynamic Braking
Detection Tracking obstacles
Camera
Braking
Navigation Stack Actuator
3D Lidar
Global Local
SLAM Controller
Map Planner Planner Drive
IMU
Motor
GNSS
Vehicle Steering
Position and Motor
Orientation
Odometry
Fig. 2: A simplified system architecture highlighting our vehicle’s key components for locomotion.
RRT [9] has been chosen as a suitable candidate to tackle Algorithm 1 Extend Children
the global planning task. The expanding tree is able to 1: children = {}
quickly navigate around the non-convex map and it presents 2: range = max_steering_angle − min_steering_angle
a trade-off between exploration and exploitation with regard 3: for i ≤ steering_samples do
to its branches. The main idea of the algorithm is to begin 4: δ ← min_steering_angle + (i/steering_samples) ·
at a starting configuration and expand that starting state in a range
tree like fashion by expanding edges of the tree inside the 5: child ← parent
unoccupied space. At its core, the algorithm consists of two 6: for j ≤ step_size do
main phases: the sampling phase and the extension phase. 7: child.x ← child.x + v · cos(child.θ ) · T
In the sampling phase, a point q_rand is randomly sam- 8: child.y ← child.y + v · sin(child.θ ) · T
pled inside the unoccupied area of the map. After that, the 9: child.θ ← child.θ + v/L · tan(δ ) · T
state q_nearest is selected. This is illustrated in the upper 10: j ← j + integration_step_size
part of Figure 3. In the extension phase, the selected state 11: children.insert(child)
is expanded in the direction of the randomly sampled point. 12: i ← i+1
This process is then repeated.
However, the vehicle in question adheres to non-
holonomic single-track kinematic constraints, meaning that
not all states that can be extended from the nearest state egy is defined in Algorithm 1 and illustrated in Figure 3.
q_nearest are valid points. Therefore, the extend strategy Essentially, the non-holonomic constraints can be imagined
will have to be modified to incorporate non-holonomic as a cone of reachable states in front of the vehicle.
constraints [18], as defined by Equations 2 – 5, where xk Further modifications and improvements to the algorithm
is the state of the vehicle at step k and uk is the control have been implemented. This paper introduces the concept
input. The state vector x is defined as (x, y, θ ), with x and y of extended dead-end nodes, which are nodes that cannot
given in m and θ ∈ (−π, π] given in rad. The control input be further expanded. An example of these extended dead-
vector u is defined as (v, δ ), where v is the linear velocity end nodes can be found in Figure 4. The idea is as follows:
of the vehicle given in m/s and δ is the front wheel steering If a node is unable to spawn any further child nodes and
angle in rad. L is the length of the vehicle and T is the all existing child nodes are marked as dead-end nodes,
discretized time step. then that parent-node is also marked as a dead-end node.
Nodes marked as dead-end are not considered for extension
xk = f(xk−1 , uk ) (2) when searching the tree for q_nearest and therefore yield a
xk = xk−1 + vk · T · cos θk−1 (3) speedup to the searching process. To the best of the authors’
knowledge, this is the first time that such extension is applied
yk = yk−1 + vk · T · sin θk−1 (4) to speed up the searching process in RRT path planners.
tan(δk ) Observation: Due to the fact that the map is finite in
θk = θk−1 + vk · T · (5)
L size and a node can only be extended into a predefined
We utilize an approach similar to [19] in order to incorpo- constant number of child-nodes, the maximum number of
rate the non-holonomic constraints into the RRT algorithm. possible nodes inside the tree is finite. Moreover, since in
To achieve this, the extend phase will be modified such that a every iteration of the algorithm, a node is either added or
constant number of child-nodes is defined, that are generated removed (marked as dead-end) from the tree, an interesting
using the non-holonomic equations, each with a steering consequence emerges: Given a starting pose x0 , the algorithm
angle δ ∈ [min_steering_angle, max_steering_angle] dis- is guaranteed to cycle through every possible tree-state
tributed evenly across the segment. The full extension strat- reachable from the starting position. This observation holdsq_nearest q_nearest q_dead_end
q_rand
q_init
q_nearest_child Fig. 4: If a parent-node is unable to spawn child-nodes (left),
q_nearest that parent is marked as a dead-end node (right).
q_rand
builds on their ideas and takes advantage of their mathemat-
ical formulations.
q_init
The vehicle model is described by non-linear time-
invariant differential equations with time step k ∈ N0 , the
state trajectory x : R 7→ X and control trajectory u : R 7→ U ,
Fig. 3: Non-holonomic RRT Algorithm: (top) Generating as given by Equations 2 – 5. The mapping f : X × U 7→ R p
random point q_rand and finding the nearest node q_nearest with p = dim(X ) defines the change of state depending
inside the tree. (bottom) The nearest node is expanded into on control input, where X and U represent the state and
child nodes out of which the nearest is found and added to control space respectively. The system is subjected to state
the tree. and input constraints originating not only from the vehicle’s
physical limitations but also from the dynamic environment
in which it drives. State constraints and input constraints will
be introduced in the following.
true regardless of the sampling strategy, even when we are
sampling from the same point in every iteration. The planning task is to guide the vehicle from xk ∈ X
at time step k = 0 to an intermediate or ultimate goal set
B. LOCAL PLANNER X f (tN ) ∈ X within the planning horizon T = N∆t while
minimizing an objective function and adhering to constraints.
The task of the local planner is to compute a sequence With the running cost γ : X × U 7→ R+ 0 and terminal
of control signals u that the vehicle has to follow. Besides cost Γ f : X 7→ R+ , the optimal control problem is given
0
knowledge of the map, the local planner also needs infor- in Equation 7. For the intents and purposes of the local
mation about the dynamic obstacles inside the environment planner, the running cost will take into account factors such
which the vehicle needs to avoid in order to not crash. The as trajectory length and distance from obstacles, whereas the
trajectory, as generated by the local planner, can be seen in terminal cost evaluates the final state xN and its distance from
Equation 6: the target point inside the global path.
N−1
v0 v1 v2 vn−1
u = δ0 , δ1 , δ2 , ..., δn−1 (6)
min
u
∑ γ(xk , uk )∆t + Γ f (xN ) (7)
k=0
∆t0 ∆t1 ∆t2 ∆tn−1
The trajectory is a sequence of commands containing subject to:
a linear velocity v and a steering angle δ . Furthermore, x0 = xs xk+1 = f(xk , uk )
temporal information is embedded inside the trajectory as xk ∈ X(k∆t) uk ∈ U(xk , k∆t)
∆t, which represents the duration of the given control input.
T = N∆t k = 0, 1, 2, ..., N − 1
The sum of times T = ∑i ∆ti is called the planning horizon
and represents how far the algorithm will plan into the future. The constraint on the state xk ∈ X(k∆t) implies that
However, here the planner only returns and executes the first the vehicle should not find itself in an illegal state, for
command of the sequence, i.e. u = [v0 , δ0 ]T in each planning example colliding with other vehicles or static obstacles. The
iteration. This in turn means that each command is then constraint is enforced as a penalty term inside the running
applied for a total time of ∆t0 = ∆t1 = ∆t2 = ... = ∆tn−1 = cost function γ(·), but its validity is also manually checked
∆t = 1/ f where f is the constant frequency with which the once a trajectory has been chosen by limiting the obstacle
planner is called. costs to a maximum tolerated value. If that value is exceeded,
Due to its ability to tackle optimization problems on a the trajectory is rendered invalid.
finite time-horizon, MPC has been chosen as a method for The constraint on the control inputs uk ∈ U(xk , k∆t) is de-
trajectory generation. The approach was inspired by [15], rived from the physical real-world limitations of the vehicleand can be mathematically formulated as: The distance to the obstacle is then passed to a function
φ (·) : R+
0 7→ R, which serves a similar purpose to the inflation
0 ≤ vk ≤ max_velocity function of the static obstacles and allows for adjusting the
(8)
min_steering_angle ≤ δk ≤ max_steering_angle shape and intensity of obstacle inflation. The simplest way
to define the function would be φ (x) = 1x . This way, a cost
Since the aim of the local planner is to follow points on gradient is introduced around each obstacle that penalizes
the global plan, a reasonable way to define the terminal cost closeness. This serves as a naïve first implementation and can
function is to define a target point xtarget on the global path be later adapted to suit the planning task best. For example, it
and penalize the final state of the trajectory based on its makes sense to also incorporate a sense of motion prediction
distance from the target point. The terminal cost is therefore inside the function in order for the planner to anticipate
defined as shown in Equation 9 where c is a constant scaling movements of the obstacles.
factor for the penalization. In addition to these, further soft constraints can be im-
posed, such as penalization on spikes and jumps in the first
Γ f (xN ) = c · kxtarget − xN k22 (9) derivative of the control input u̇. For instance, introducing
This definition of the terminal cost function makes in- terms such as kuk+1 − uk k1 penalizes variations in control
tuitive sense, especially considering that the target points inputs and yields a smoother trajectory.
xtarget moves along the global path as soon as the vehicle The next section deals with the evaluation of the suggested
is within a certain range of it. This range is defined as the planning method.
global_lookahead_distance. On the other hand, defining the V. EVALUATION
running cost γ(·) on the other hand, involves three different
The evaluation of the approach is done in two stages,
terms and can be seen in Equation 10:
the first stage being inside a simulation environment and
the second stage in the form of live tests on the vehicle.
γ(xk , uk ) = cl · vk · ∆t + cm · m(xk , yk ) + co · o(xk ) (10) The optimal control problem is formulated and solved using
The terms cl , cm and co represent cost scaling factors and Google’s Ceres Solver [20].
are of high importance for the planning process. Namely, the We conduct the simulation inside the Stage simulator ROS
cost factors directly influence how much each of the penalty adaptation StageROS8 . The purpose of the simulations is
terms in the cost function influences the overall cost and to tune the planning parameters and evaluate the planner
therefore changes the shape of the final trajectory. based on its ability to follow target points on a global path
The first term of the running cost penalizes the length of while avoiding static and dynamic obstacles. The dynamic
the trajectory. This is done by penalizing the velocity as a obstacles represent pedestrians inside the map, which are
higher velocity yields a longer trajectory. simulated using the PedSim9 simulator which uses the social
The next term is penalization with respect to the map. force model for pedestrian dynamics [21].
The function m(·) evaluates the state based on the state The simulations have shown to be successful, with the
coordinates (xk , yk ) and returns a discrete cost value between vehicle reaching its target position while also managing to
[0, 255] where 0 means that the space is free, and 255 implies avoid dynamic obstacles. The global planner was able to
that the state is occupied. This value gradually decreases generate paths of 150 m length in 0.2 s of computation time,
based on the distance from the obstacle, usually given by an whereas the local planner operates at a frequency of 5 Hz
inflation function. This inflation directly introduces a gradi- with a planning horizon of up to T = 12 s at times. These
ent that pushes the trajectory further away from the obstacle, results have permitted to move ahead and test the approach
by increasing running costs. The inflation is generated by the on the target vehicle.
ROS map server package7 provided in the navigation stack. Over the period of a few months, several tests have been
The final term penalizes the state based on its relation to conducted live on the vehicle, with the most important
dynamic obstacles. This is done in a similar fashion to the improvements and highlights being recorded and uploaded to
way the previous term was evaluated. However, this term also YouTube. In the real life tests, the planner has demonstrated
offers the possibility of further extending the cost function its ability to calculate and follow a global path and reach the
with dynamic obstacle motion prediction. Equation 11 offers designated goal area while driving at a maximum speed of
a simple way of evaluating that problem, where M is the total 1 m/s. Due to the inflation of the static obstacles inside the
number of obstacles. map, the vehicle was able to keep a safe distance from walls
M
and stay inside the designated driving area10 . This holds true
o(xk ) = ∑ φ (kobsi (k∆t) − xk k2 ) (11) even when driving at night time11 .
i=1 After that, further tests with dynamic obstacles were con-
The obstacle position is estimated at the time k∆t and the ducted. The vehicle is able to correctly identify pedestrians
distance is evaluated and penalized. The term obsi (k∆t) is 8 http://wiki.ros.org/stage_ros
defined as the geometrical center of the obstacle i. 9 https://github.com/srl-freiburg/pedsim_ros
10 https://youtu.be/grg-59TbCVY
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12 https://youtu.be/Rnf0l_RLoyoYou can also read
